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On the dynamical nature of Saturn’s North Polar hexagon
 Masoud Rostami, (mrostami@lmd.ens.fr), Vladimir Zeitlin , Aymeric Spiga

This is the Authors’ Original Manuscript (AOM); that is, the manuscript in its
original form; a “preprint”.
Published online in the ICARUS: 19 June 2017
https://doi.org/10.1016/j.icarus.2017.06.006
DOI: 10.1016/j.icarus.2017.06.006

On the dynamical nature of Saturn’s North
Polar hexagon.
Masoud Rostamia , Vladimir Zeitlinb ? , and Aymeric Spigac
a,b,c Laboratoire

de M´et´eorologie Dynamique (LMD)/IPSL, Universit´e Pierre et
Marie Curie (UPMC), Paris, France

a,b Ecole

Normale Sup´erieure(ENS), Paris, France

ABSTRACT
An explanation of long-lived Saturn’s North Polar hexagonal circumpolar jet
in terms of instability of the coupled system polar vortex - circumpolar jet is
proposed in the framework of the rotating shallow water model, where
scarcely known vertical structure of the Saturn’s atmosphere is averaged out.
The absence of a hexagonal structure at Saturn’s South Pole is explained
similarly. By using the latest state-of-the-art observed winds in Saturn’s
polar regions a detailed linear stability analysis of the circumpolar jet is
performed (i) excluding (“jet-only” configuration), and (2) including
(“jet+vortex” configuration) the north polar vortex in the system. A domain
of parameters: latitude of the circumpolar jet and curvature of its azimuthal
velocity profile, where the most unstable mode of the system has azimuthal
wavenumber 6, is identified. Fully nonlinear simulations are then performed,
initialized either with the most unstable mode of small amplitude, or with
the random combination of unstable modes. It is shown that developing
barotropic instability of the “jet+vortex” system produces a long-living
structure akin to the observed hexagon, which is not the case of the
“jet-only” system, which was studied in this context in a number of papers
in literature. The north polar vortex, thus, plays a decisive dynamical role.
The influence of moist convection, which was recently suggested to be at the
origin of Saturn’s north polar vortex system in the literature, is investigated
in the framework of the model and does not alter the conclusions.

Keywords: Saturn’s Hexagon ; Barotropic Instability ; Rotating Shallow
Water Model
? Corresponding author. Email: zeitlin@lmd.ens.fr Address: LMD-ENS, 24 Rue
Lhomond, 75005 Paris, France

Preprint submitted to Icarus

https://doi.org/10.1016/j.icarus.2017.06.006
May 11, 2017
DOI: 10.1016/j.icarus.2017.06.006

1

Introduction

Visible imagery acquired during the Voyager 1 & 2 fly-by of Saturn in 1980-1981
revealed the hexagonal cloud pattern of Saturn’s north pole at about 77◦ N planetographic latitude (Godfrey, 1988). As followed from these early observations, the
motion of the associated cloud structures suggested that the structure is related to
a circumpolar jet-stream, but the hexagon pattern itself appeared to be stationary
(i.e. very close to the rotation of Saturn’s interior). The early diagnostics were
confirmed first in the 1990s by Hubble Space Telescope observations (SanchezLavega et al., 1993), and then in the 2000s by the Cassini orbiter firstly in thermal
infrared images (Baines et al., 2009), followed by visible imagery once Saturn’s
northern pole came out of the winter darkness. The high spatial resolution of the
Cassini visible and infrared images permitted to evidence the presence of the north
polar vortex (NPV), in addition to the hexagonal jet. The exceptional duration
of the Cassini mission allowed S´anchez-Lavega et al. (2014) to conclude that the
hexagon resists seasonal changes, before Antunano et al. (2015) compiled the repeated cloud-layer observations of the hexagon to obtain a complete profiling of
the winds in this structure. Thus far, such hexagonal feature has not been observed
at Saturn’s South pole.
The persisting hexagonal pattern in polar projected maps demonstrates that a
prominent wavenumber 6 perturbation shapes Saturn’s circumpolar jet stream at
latitude ≈77◦ N. An early interpretation by Allison et al. (1990) was that the anticyclonic North Polar Spot (NPS) vortex (not to confuse with the polar vortex),
visible at that time in the vicinity of the hexagon forced a stationary planetary
(Rossby) wave which gave the polar jet its hexagonal shape, but the apparent
disappearance of the NPS between 1995 and 2004 invalidated this scenario (see
Sayanagi et al. (2016) for the history of the observations of the NPS and the
hexagon). The nature and origin of the hexagon has been addressed since then
both in laboratory experiments and numerical models. Polygonal structures are
often reported in rotating flow experiments, although the wavenumber 6 configuration remained elusive (Vatistas et al., 1994; Marcus and Lee, 1998; Jansson
et al., 2006; Bergmann et al., 2011). Aguilar et al. (2010) discussed more specifically the relevance of rotating tank experiments to modeling Saturn’s polar jet
and concluded that the hexagonal structure could be resulting from the barotropic
instability of the jet, and that the wavenumber 6 prominence is conditioned by the
intensity of the jet and bottom friction. They were first to notice a dependence
2

of the azimuthal wavenumber of the most unstable mode on the deformation radius in their instability analysis in the framework of the quasi-geostrophic (QG)
model. Using a general circulation model with a domain limited to Saturn’s northern polar region, and a polar jet as initial condition, Morales-Juberias et al. (2011)
reproduced the structures found by Aguilar et al. (2010) in laboratory experiments: the hexagonal shape resulting from a vortex street formed by developing
barotropic instability of the jet, with cyclonic and anticyclonic vortices at poleward
and equator-ward sides of the jet, respectively. Yet, the vortices forming the street
were too large and too strong, with respect to observations, and this mechanism
was giving rise to larger propagation speed than the observed one (S´anchez-Lavega
et al., 2014). Morales-Juberias et al. (2015) proposed an alternative “meandering
jet” model which matches the morphology (including sharp potential vorticity,
hereafter PV, fronts) and phase speed of Saturn’s hexagon, provided that an ad
hoc vertical shear of the jet is introduced. The same authors concluded that deep
jets evolve into vortex streets and shallow jets evolve into meanders, both being
possibly at the origin of the absence of seasonal variability of the hexagonal jet.
Although the meandering jet model of Morales-Juberias et al. (2015) offers thus
far the best agreement with the characteristics of the observed Saturn’s hexagon,
the putative meridional temperature gradients accompanying the vertical shear of
the (supposedly shallow) polar jet are yet to be confirmed.
Despite these developments, there is not enough convincing dynamical explanation of the existence and origin of Saturn’s North polar hexagon, as well as the
absence of its counterpart at Saturn’s South pole. In the present paper we propose an alternative approach to the problem. Our analysis is performed in the
framework of a simple rotating shallow water (RSW) model, which is of use for
modeling atmospheres of giant planets, cf. (Dowling, 1995). A specificity of our
approach is that the model is considered in the polar tangent plane approximation, the so-called gamma-plane, in order to take into account the variations of the
Coriolis parameter with latitude in the polar regions. The use of the model, which
may be obtained by vertical averaging of the full primitive equations, is justified
by the fact that information on the vertical structure of Saturn’s atmosphere is
scarce. All vertical structure being averaged out, the model is barotropic, in the
sense that it does not contain any vertical shear, although it does allow for vertical
stretching of fluid columns. In the same sense the instabilities in the model will
be called barotropic below. The model combines simplicity with dynamical consistency and, with the astronomical parameters of Saturn being given, contains a
single adjustable parameter: the effective Rossby deformation radius, Rd . The QG
model used for theoretical analysis in Aguilar et al. (2010) can be obtained from
RSW in the limit of vanishing Rossby numbers. The RSW model, which allows for
efficient high-resolution numerical implementations, has been used for the analysis
of various features of the Saturn’s atmosphere: mid-latitude jets (Showman, 2007),

3

equatorial super-rotating jet (Scott and Polvani, 2008), NPV (O’Neill et al., 2016)
(in two-layer version with a moist-convective forcing in the latter work). Yet, to our
knowledge, there are no studies in the literature applying the RSW model to the
developing barotropic instability of the circumpolar Saturn’s jet. The particularity
of our approach is that we explore the dynamics in two different configurations
of the mean zonal velocity profile: (i) considering only the 77◦ N circumpolar jet,
with observed velocity profile, and no central vortex at 90◦ N (i.e. the traditional
“jet-only” configuration considered in all above-cited papers), and (ii) considering
the full “jet + vortex” system with the observed zonal velocity profile, which was
not considered in the existing literature in this context. The state-of-the-art mean
zonal velocity profiles on Saturn (Antunano et al., 2015) are used in linear stability
studies and for initializations of nonlinear simulations.
The paper is organized as follows: in section 2 we introduce the model, discuss its
general features and vortex solutions, present the analytical fits of the observed
velocity profiles, and formulate the linear stability problem. Section 3 is devoted to
the study of the jet-only configuration, with presentation of the results of the linear
stability analysis and nonlinear simulations of the saturation of the instability. In
section 4 we analyse the jet+vortex configuration along the same lines. Finally, in
section 5 we present our conclusions and discussion. A discussion of the influence
of the effects of moist convection upon the evolution of the instability is given in
the Appendix.

2

2.1

Rotating shallow water model for large-scale atmospheric motions,
vortex solutions and their (in)stability

The model

Rotating shallow water equations, in the absence of forcing and dissipation read:
v
Dv
+ (f + )(ˆ
z × v ) = −g∇h,
Dt
r

(2.1)

Dh
+ h∇.v = 0.
(2.2)
Dt
Here, by anticipating the application to polar jets and vortices, we use the cylindrical coordinates on the plane (r, θ), and zˆ is the unit vector normal to the plane.
v = uˆ
r + v θˆ is the horizontal velocity with u, v denoting radial and azimuthal
4

components, respectively, D/Dt = ∂/∂t + v · ∇ denotes the material derivative,
ˆ
∇ = rˆ∂r + θ(1/r)∂
θ is the gradient in the plane, where ∂r and ∂θ denote partial
derivatives with respect to corresponding arguments. h is the thickness of the layer,
g is gravity and f is the Coriolis parameter.
Bottom topography b(r, θ) can be easily introduced in the model by replacing h by
h − b in (2.2), and can be used to parameterize the deep layers of the atmosphere.
This would however introduce ad hoc parameters, which we want to avoid.
The RSW equations possess a Lagrangian invariant, the potential vorticity (PV):
q=

zˆ · (∇ × v ) + f
,
h

(2.3)

where zˆ · (∇ × v ) is relative vorticity. The PV anomaly
P V A = q − f0 /H0 ,

(2.4)

where f0 and H0 are reference values of Coriolis parameter and thickness of the
layer (see below), will be used for diagnostics in what follows. We consider the
RSW equations in the polar tangent plane, the so-called γ-plane, where the effects
of sphericity in the Coriolis parameter are taken into account in the lowest-order
approximation:
!

Rs
.
Rd
(2.5)
Here f0 = 2 Ωs , φ is planetographic latitude, Ωs and Rs represent Saturn’s angular
velocity and radius, respectively (we will neglect the effects of non-sphericity in
what follows). Approximate values for the mean Saturn radius and f0 are respectively 55000 km and 3.2 × 10−4 s−1 (Read et al., 2009b; S´anchez-Lavega et al.,
2014). As is well-known, the rotating shallow
√ water system possesses an internal
scale, the Rossby deformation radius Rd = gH0 /f0 .
f = f0 sin(φ) = f0



1 2
∗ ∗2
+ ... ;
1−
2 r + ... = f0 1 − γ r
2Rs

0 ≤ r∗ ≤

Equations (2.1), (2.2) can be obtained by vertical averaging between two material
surfaces of the full three-dimensional primitive equations for the atmosphere in
pseudo-height isobaric coordinates, e.g. (Bouchut et al., 2009), when one of these
surfaces is considered fixed (a constant pressure level), and one is free. The potential temperature is considered uniform through the layer in this approximation,
although an extension of the model including horizontal potential temperature
gradients is possible. In this case g is the gravity acceleration of Saturn, and H0 is
the thickness of the layer. The model can be also considered as a two-layer model
with strong disparity in depths and/or densities of the layers. In this case H0 plays
5

a role of equivalent depth, and g is so-called reduced gravity, i.e. gravity weighted
with the stratification parameter. The deformation radius in this interpretation
is related to stratification properties of the atmosphere. Whatever the interpretation is, at a given rotation rate f0 /2 , the deformation radius Rd√is the only free
parameter of the model, as the velocity scale can be taken to be gH0 .
In terms of velocity components, the momentum and mass conservation equations
of the model read:

2.2

Du v 2

− fv = −g∂r h,
Dt
r

(2.6)

Dv uv
+
+ fu = −g∂θ h,
Dt
r

(2.7)

1
1
∂t h + ∂r (hru) + ∂θ (hv) = 0.
r
r

(2.8)

Vortex solutions

Stationary axisymmetric vortex solutions of the equations (2.6) - (2.8) are those
with zero radial velocity u = 0, some axisymmetric distribution of azimuthal velocity v = V (r), and the thickness field H(r) related to V (r) through the cyclogeostrophic (gradient wind) balance:
V2
+ f V = g∂r H.
r

(2.9)

For a given V (r), corresponding thickness profile H(r) can be obtained by integration of equation (2.9). Note that in the QG model used for theoretical analysis
in Aguilar et al. (2010) the centrifugal acceleration (the first term in (2.9)) is neglected. While this approximation is fully justified for the circumpolar jet, it is
less so for the central vortex. Herein we consider solutions of (2.9) with V (r) corresponding to the observed time- and zonally averaged profiles of Saturn’s zonal
wind (Antunano et al., 2015). These profiles, with error margins, are represented
by dashed lines in Figure 1.
It is useful to have a simple analytic expression mimicking the observed velocity
profiles. Although the linear stability analysis can be accomplished directly with
digitized observational velocity profile, it is convenient to control the shape of
the velocity profile in terms of a small number of parameters. Dependence on
6

0.16

0.16

0.14

0.14

0.12

0.12
0.1
Velocity

Velocity

0.1
0.08
0.06

0.08
0.06

0.04

0.04

0.02

0.02

0

0

0

1

2

3

4

5

6

0

1

2

3

r

4

5

6

r

3

3

2.5

2.5

2

2

PV

PV

Figure 1. Observed zonal velocity profiles in the northern (left panel) and southern
(right panel) hemispheres and their fits used in the paper. Three dashed lines represent
the observed non-dimensional mean and error margins of the averaged zonal velocity
(Antunano et al., 2015), and the solid line is the composite of two model profiles (in
red) fitting the NPV and the circumpolar jet. The fit of the velocity profile of the
NPV gives = 0.15, α = 0.42, β = 1.3, r0 = 0, m = 1 and that of the circumpolar jet
gives = 0.08, α = 0, β = 2, r0 = 3.37, m = 3. Similarly, south polar vortex correspond to = 0.16, α = 0.5, β = 1.2, r0 = 0, m = 1 and southern circumpolar jet jet to
= 0.08, α = 0, β = 2, r0 = 4.6, m = 3. Rd ≈ 3200 km is taken for both poles.

1.5

1.5

1

1

0

1

2

3

4

5

6

0

r

1

2

3

4

5

6

r

Figure 2. PV distributions corresponding to observed velocity profiles of Fig. 1 in the
northern (left panel) and southern (right panel) hemispheres and their analytic fits.

these parameters could be then analysed. The PV, which is a key characteristic of
the flow, is very noisy, if determined directly from the observed profiles, cf. Fig.
2, whereas a PV field computed from the analytic fit enables a straightforward
analysis of possible instability, see below. For initialization of the non-linear RSW
simulations, the use of the analytic fit allows for straightforward integration of
(2.9), and thus diminish the discretisation errors. Therefore, we fit the peaks of
7

the velocity distribution with the help of the following simple formula:
β

V (r) = (r − r0 )α e−m(r−r0 ) ,

α, β, , m ≥ 0,

(2.10)

where measures the intensity of the velocity field, r0 tunes the distance of the
velocity peak of the jet from the pole, and other parameters allow to fit the shape
of the distribution (r0 = 0 for the central vortex and α = 0 for the jet). All velocity
profiles are normalized by their maximum values, so the maximum value of velocity
is equal to . By applying this formula to both central vortex and circumpolar jet
(red curves in Fig. 1), we get synthetic profiles represented by solid black lines in
the figure. We do not seek to reproduce the “shoulder” in the velocity profile of
the central vortex, as it turns out that it is inessential for the dominant long-wave
instability of the system which we are interested in (a test study was performed
to check this fact).
The stability of any velocity profile V (r) (and corresponding H(r) obtained from
the cyclo-geostrophic balance) with respect to small perturbations can be analyzed
by standard means. We represent each field as the solution in question and a small
perturbation (denoted by a prime):
u(r, θ, t) = u0 (r, θ, t),

(2.11)

v(r, θ, t) = V (r) + v 0 (r, θ, t),

(2.12)

h(r, θ, t) = H(r) + h0 (r, θ, t),

(2.13)

and inject these expressions in the full system (2.6) - (2.8). Retaining only the linear
contributions in perturbation terms, we are looking for normal-mode solutions with
harmonic dependence on time and polar angle
i(lθ−ωt)
˜
(u0 , v 0 , h0 )(r, θ, t) = Re[(i˜
u, v˜, h)(r)e
],

(2.14)

where l and ω are the azimuthal wavenumber and the complex eigenfrequency,
ω = ωR + iωI . We thus formulate the problem as an eigenvalue problem for eigenfrequencies ω. A positive imaginary part for an eigenfrequency ωI corresponds to
an instability with a linear growth rate σ = ωI .
Following common practice, it is convenient to formulate this eigenproblem in non−1
dimensional terms. By
√ using the deformation radius Rd as the scale for r, f0 as
the scale for t, and gH0 as the scale for velocity, the eigenproblem in question
8


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